Question

The length of a metal wire is $Z_1$, when the longitudinal strain in it is $T_1$ and when the longitudinal strain be $T_2$ then length of wire is $Z_2$ . The unstretched length of the wire is

• A. $\sqrt{Z_1{Z}_2}$
  
• B. $\frac{Z_1+Z_2}{2}$
  
• C. $\frac{T_1{Z}_2-T_2{Z}_1}{T_1-T_2}$
  
• D. $\frac{Z_1{T}_2+Z_2{T}_1}{T_2+T_1}$
  

Answer 1

The correct option is C $\frac{T_1{Z}_2-T_2{Z}_1}{T_1-T_2}$


$\text{Longitudinal strain}=\frac{\text{Change in length}}{\text{Original length}}=\frac{\Delta Z}{Z}$
$\Rightarrow$ when length of wire is $Z_2$ will be,
$\Delta Z=Z_2-Z$
$\Rightarrow$ when length of wire is $Z_1$ will be,
$\Delta Z=Z_1-Z$

Where $Z$ is original/unstretched length.
$\therefore T_1=\frac{\Delta Z}{Z}=\frac{Z_1-Z}{Z}...\left(i\right)$
and
$T_2=\frac{\Delta Z}{Z}=\frac{Z_2-Z}{Z}...\left(ii\right)$
Dividing equation $\left(i\right)$ and $\left(ii\right)$ gives:

$\frac{T_1}{T_2}=\frac{Z_1-Z}{Z_2-Z}$

$\therefore Z=\frac{T_1{Z}_2-T_2{Z}_1}{T_1-T_2}$